$\gamma \gamma \to {\pi }^0{\pi }^0$ and $K_L\to {\pi }^0\gamma \gamma$ in the chiral quark model

We consider the process $\gamma \gamma \to {\pi }^0{\pi }^0$ within the quark model. We find small corrections to the lowest-order result obtained from chiral loops. These corrections are very similar to those found in vector-meson-dominated models, and too small to be tested with the present data. We use this result, together with a simple pole model, to find the amplitude for $K_L\to {\pi }^0\gamma \gamma$ and use this to estimate the $\mathrm{CP}$-conserving part of $K_L\to {\pi }^0{e}^{+}{e}^{-}$. For this last process we find a branching ratio ∼2.3×${10}^{-13\mathrm{}}$, significantly smaller than the $\mathrm{CP}$-violating contribution.